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National and Regional Contests
Puerto Rico Contests
Puerto Rico Team Selection Test
2009 Puerto Rico Team Selection Test
2009 Puerto Rico Team Selection Test
Part of
Puerto Rico Team Selection Test
Subcontests
(6)
6
1
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for which n can the nxn board be colored black in a finite number of steps
The entries on an
n
n
n
×
n
n
n
board are colored black and white like it is usually done in a chessboard, and the upper left hand corner is black.We color the entries on the chess board black according to the following rule: In each step we choose an arbitrary
2
2
2
×
3
3
3
or
3
3
3
×
2
2
2
rectangle that still contains
3
3
3
white entries, and we color these three entries black.For which values of
n
n
n
can the whole board be colored black in a finite number of steps
5
2
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computational with a cyclic ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral inscribed in a circle. The diagonal
B
D
BD
B
D
bisects
A
C
AC
A
C
. If
A
B
=
10
AB = 10
A
B
=
10
,
A
D
=
12
AD = 12
A
D
=
12
and
D
C
=
11
DC = 11
D
C
=
11
, find
B
C
BC
BC
.
2009 integers around a circle, weird mean Puerto Rico Ibero IMO TST 2009.5
The weird mean of two numbers
a
a
a
and
b
b
b
is defined as
2
a
2
+
3
b
2
5
\sqrt {\frac {2a^2 + 3b^2}{5}}
5
2
a
2
+
3
b
2
.
2009
2009
2009
positive integers are placed around a circle such that each number is equal to the the weird mean of the two numbers beside it. Show that these
2009
2009
2009
numbers must be equal.
4
2
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x_1^2 + x_2^2 = 5 if x^2 - bx + c = 0 has 2 real roots
Find all integers
b
b
b
and
c
c
c
such that the equation
x
2
−
b
x
+
c
=
0
x^2 - bx + c = 0
x
2
−
b
x
+
c
=
0
has two real roots
x
1
,
x
2
x_1, x_2
x
1
,
x
2
satisfying
x
1
2
+
x
2
2
=
5
x_1^2 + x_2^2 = 5
x
1
2
+
x
2
2
=
5
.
<MAB=<MCB iff <MBA=<MDA, ABCD # Puerto Rico Ibero IMO TST 2009.4
The point
M
M
M
is chosen inside parallelogram
A
B
C
D
ABCD
A
BC
D
. Show that
∠
M
A
B
\angle MAB
∠
M
A
B
is congruent to
∠
M
C
B
\angle MCB
∠
MCB
, if and only if
∠
M
B
A
\angle MBA
∠
MB
A
and
∠
M
D
A
\angle MDA
∠
M
D
A
are congruent.
3
2
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(AC)^2 - (CE)^2 = (AB)^2 - (EB)^2
On an arbitrary triangle
A
B
C
ABC
A
BC
let
E
E
E
be a point on the height from
A
A
A
. Prove that
(
A
C
)
2
−
(
C
E
)
2
=
(
A
B
)
2
−
(
E
B
)
2
(AC)^2 - (CE)^2 = (AB)^2 - (EB)^2
(
A
C
)
2
−
(
CE
)
2
=
(
A
B
)
2
−
(
EB
)
2
.
h_A + h_B + h_C >= 9r Puerto Rico Ibero IMO TST 2009.3
Show that if
h
A
,
h
B
,
h_A, h_B,
h
A
,
h
B
,
and
h
C
h_C
h
C
are the altitudes of
△
A
B
C
\triangle ABC
△
A
BC
, and
r
r
r
is the radius of the incircle, then
h
A
+
h
B
+
h
C
≥
9
r
h_A + h_B + h_C \ge 9r
h
A
+
h
B
+
h
C
≥
9
r
2
2
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N perfect square if last three digits of N are x25,
The last three digits of
N
N
N
are
x
25
x25
x
25
. For how many values of
x
x
x
can
N
N
N
be the square of an integer?
0 or 1 inside 2009 boxes Puerto Rico Ibero IMO TST 2009.2
In each box of a
1
×
2009
1 \times 2009
1
×
2009
grid, we place either a
0
0
0
or a
1
1
1
, such that the sum of any
90
90
90
consecutive boxes is
65
65
65
. Determine all possible values of the sum of the
2009
2009
2009
boxes in the grid.
1
2
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Handshakes
By the time a party is over,
28
28
28
handshakes have occurred. If everyone shook everyone else's hand once, how many people attended the party?
5 is good, 2005 is better, and 2005^2 is best Puerto Rico Ibero IMO TST 2009.1
A positive integer is called good if it can be written as the sum of two distinct integer squares. A positive integer is called better if it can be written in at least two was as the sum of two integer squares. A positive integer is called best if it can be written in at least four ways as the sum of two distinct integer squares. a) Prove that the product of two good numbers is good. b) Prove that
5
5
5
is good,
2005
2005
2005
is better, and
200
5
2
2005^2
200
5
2
is best.